Now, imagine that we clear the space around the black hole and use some kind of spaceship to first build up a near light speed velocity. Then, from a distance of a few "light days" (we suppose that this black hole has a diameter of almost a "light day") we aim straight at the black hole, turn off the engines and just plainly falls towards it.

My question is; If the spaceship already is at very near light speed, why doesn't the massive gravitational field from the black hole manage to accelerate the spaceship to above light speed before it reaches the black hole? Any relativistic mass increase of the spaceship would not make any difference since it is accelerated by a gravitational field (and not by its engines), so what is stopping it?

Some thought needs to be given as to how the velocity is being measured. For instance, the coordiante velocity dr/dt for an object falling into a black hole will be zero! The most convenient number is not a velocity, but a rapidity - dr/dtau, which is the rate of change of the Schwarzschild coordinate 'r' with the proper time 'tau'.